To answer your question, we can use real life situations. Say you have one apple, and I also have one apple. In total, how many apples do we have? My one, plus your one, is two... Assuming we are not counting any "half-apples" and so on and so forth...
I like to think that addition is only defined, not proven. The "belief" that one plus one equals two is merely an axiom. If "two" were called anything else, wouldn't the equation still be correct, even though it's disputed that 1 + 1 can only = 2 in this reality? Two is accepted to be the sum of one and one.